Related papers: A General Reciprocity Law for Symbols on Arbitrary…
Lorentz's reciprocity lemma and Feld-Tai reciprocity theorem show the effect of interchanging the action and reaction in Maxwell's equations. We derive a free-space version of these reciprocity relations which generalizes the conservation…
Over an algebraically closed field, the $\textit{double point interpolation}$ problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points. After being open for 90 years, a…
We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the…
The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of…
We introduce the notions of Collective Arbitrage and of Collective Super-replication in a discrete-time setting where agents are investing in their markets and are allowed to cooperate through exchanges. We accordingly establish versions of…
In this paper, using what we call a micro reciprocity law, we complete Weil's program for non-abelian class field theory of Riemann surfaces.
The latest developments in AI focus on agentic systems where artificial and human agents cooperate to realize global goals. An example is collaborative learning, which aims to train a global model based on data from individual agents. A…
We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We…
We use group representation theory to give algebraic formulae to compute complete transversals of singularities of vector fields, either in the nonsymmetric or in the reversible equivariant contexts. This computation produces normal forms…
The notion of geometric version of an infinitely divisible law is introduced. Concepts parallel to attraction and partial attraction are developed and studied in the setup of geometric summing of random variables.
The paper describes an explicit combinatorial formula for a harmonic vector for the Laplacian of a directed graph with arbitrary edge weights. This result was motivated by questions from mathematical economics, and the formula plays a…
Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…
The aim of this paper is to establish the Marcinkiewicz-Zygmund (MZ) type law of large numbers for the randomly weighted sums with weights chosen randomly, uniformly over the unit sphere in $\mathbb{R}^n$. We also establish a theorem that…
In this paper, we establish a new law of large numbers with the rate of convergence for special partial sums in a probability space. The proof relies on nonlinear expectation theory, as the uncertainty of random variables in the special…
The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often…
It is proposed that to the usual probability theory, three definitions and a new theorem are added, the resulting theory allows one to displace the central role usually given to the notion of conditional probability. When a mapping $\phi$…
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional…
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or…